Subj basic cal

SCENARIO:

Imagine: You are a packaging consultant for a company launching a new product. They need a box with a square base and an open top to hold exactly 100 cubic centimeters of their unique product. However, they are budget-conscious and want to minimize the cost of materials.

Your Mission: As their trusted packaging consultant, you will be responsible for designing the optimal box. What are the ideal dimensions (length and width of the base, and height) that will minimize the total amount of material used while still holding the required volume?

Think like a consultant: How can you apply your mathematical expertise to optimize the box design? Can you develop a formula to find the perfect balance between volume and material usage? Keep in mind that the client values both affordability and functionality, so your solution should be cost-effective while meeting their volume requirements.

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TASK Based on the scenario above: Develop a mathematical model using appropriate functions and constants. Apply calculus techniques (derivatives, critical points, optimization tests) to find the optimal solution. Verify your solution and ensure it adheres to the constraints.